UC-NRLF 


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Geometric  Properties  Completely  Characterizing 

the  Set  of  All  the  Curves  of  Constant 

Pressure  in  a  Field  of  Force 


'   4  I 
BY 

Eugenie  M.  Morenus 


Submitted  in  partial  fulfillment  of  the  requirements  for  the  degree  of 

Doctor  of  Philosophy  in  the  Faculty  of  Pure 

Science,  Columbia  University 


PRESS  OF 

THE  JOHN  C.  WINSTON  COMPANY 

PHILADELPHIA 

I92Z 


.         » ■     I    .     .  . 


Geometric  Properties  Completely  Characterizing 

the  Set  of  All  the  Curves  of  Constant 

Pressure  in  a  Field  of  Force 


BY 

Eugenie  M.  Morenus 


o^ 


% 


Submitted  in  partial  fulfillment  of  the  requirements  for  the  degree  of 

Doctor  of  Philosophy  in  the  Faculty  of  Pure 

Science.  Columbia  University 


PRESS  OF 

THE  JOHN  C.  WINSTON  COMPANY 

PHILADELPHIA 

1922 


CONTENTS 

PAGE 

Introduction v 

Chapter  I.    Geometric   Properties    of    the    Set    of     oo  * 

Curves  Sc 1 

Section  1.  The  differential  equation  of  Sc 1 

2.  Osculating  parabolas 2 

3.  Hyperosculating  circles 2 

4.  The  focal  circle 3 

5.  Correspondence  between  acceleration  vectors  and 

element  tangents 5 

6.  Characteristics  of  the  quartic  generated 6 

7.  The  force  vector 8 

8.  The  radius  of  the  hyperosculating  circle 8 

Chapter  II.  The  Converse  Problem  for  Sc 10 

Section  1.  Nature  of  the  Converse  Problem 10 

2.  Converse  of  Property  1 10 

3.  Converse  of  Property  2 10 

4.  Converse  of  Property  3 11 

5.  Converse  of  Property  4 15 

6.  The  parameter 16 

Chapter  III.  Geometric    Properties    of    the    Quadruply 

Infinite  System  5 18 

Section  1.  Intrinsic  equation  of  the  system 18 

2.  The  Cartesian  equation 19 

3.  Osculating  conies 20 

4.  Circles  as  curves  of  infinite  pressure 20 

5.  Hyperosculating  circles 21 

6.  The  length  of  the  force  vector 22 

7.  The  radius  of  curvature  of  conic  1 22 

8.  Second  center  of  curvature  of  conic  1 23 

9.  Intercept  of  the  asymptote  of  the  cubic  of  Property  4  24 

10.  Direction  of  the  cubic 25 

1 1 .  Hyperosculating  parabolas 26 

12.  Intercept  of  the  quintic  of  Property  7 28 

(iii) 


478713 


IV  CONTENTS. 

PAGE 

Chapter  IV.  The  Converse  Problem  for  S 29 

Section  1 .  Converse  of  Property  1 29 

2.  Converse  of  Property  2 29 

3.  Converse  of  Property  3 29 

4.  Converse  of  Property  4 31 

5.  Converse  of  Property  5 32 

6.  Outline  of  final  steps -. 33 

7.  Converse  of  Property  6 33 

8.  Converse  of  Property  7 34 

9.  Converse  of  Property  8 35 


CURVES  OF  CONSTANT  PRESSURE 


INTRODUCTION 

In  the  Princeton  Colloquium  lectures,  1909,  Professor  Edward 
Kasner  of  Columbia  University  pointed  out  several  unfinished 
problems  connected  with  a  field  of  force.  He  showed  that  the 
trajectories  whose  characteristics  he  had  previously  described 
(Transactions  of  the  American  Mathematical  Society,  Vol.  7, 
No.  3,  pp.  401-424,  July,  1906)  might  be  considered  as  a  special 
case  of  either  of  two  more  general  problems :  to  find  curves  along 
which  a  constrained  motion  is  possible  such  that  the  pressure  of 
the  moving  particle  against  the  curve  is  (1)  proportional  to  the 
normal  component  of  the  force  or  (2)  constant. 

The  pressure,  since  the  curve  is  considered  smooth,  is  con- 
nected with  the  normal  component  of  acceleration  by  the  formula 

v2 

P  = N.    In  the  case  of  trajectories  a  particle  moves  freely 

r 

under  the  action  of  a  force  which  depends  only  on  the  position  of 
the  particle;  that  is,  there  is  no  pressure  and  P  =  o.  P  =  o  is  ob- 
tained when  k  =  o  from  P  =  k  N,  which  represents  the  first  general 
problem,  or  when  c  =  o  from  P  =  c,  which  represents  the  second 
general  problem. 

Regarding  P  =  o  as  a  special  case  of  P  =  k  N,  Professor  Kasner 
stated  five  properties  characterizing  the  system  S*  of  <» 3  curves 
corresponding  to  any  value  of  the  parameter  k.  Sarah  Elizabeth 
Cronin  in  her  dissertation,  1917,  found  geometric  properties 
completely  characterizing  the  system  of  °° 4  curves  obtained  by 
combining  all  the  systems  Sjt. 

It  is  my  purpose  to  consider  the  problem  represented  by  P  =  c, 
the  problem  of  curves  along  which  a  constrained  motion  is  possible 
such  that  the  pressure  against  the  curve  remains  constant. 

I.  I  shall  prove  that  the  system  Sc  of  °o3  curves  of  constant 
pressure  corresponding  to  any  one  value  of  the  parameter  c  has 
four  properties. 

Property  1.  For  any  given  lineal  element  (x,  y,  y')  the  foci 
of  the  osculating  parabolas  of  the  single  infinity  of  curves  deter- 
mined by  the  given  element  lie  on  a  circle  passing  through  the 
given  point. 

(v) 


Vi  INTRODUCTION. 

Property  .2.  Of  the  <» l  curves  having  the  given  lineal  element 
(x,  y,  y' ,)  one  has  contact  of  the  third  order  with  its  circle  of 
curvature.  The  locus  of  centers  of  the  oo1  hyperosculating  circles 
obtained  by  varying  the  initial  direction  is  a  conic  passing  through 
the  given  point  in  the  direction  of  the  force  acting  at  that  point. 

Property  3.  The  circle  that  corresponds  according  to  Prop- 
erty 1  to  a  given  lineal  element  is  so  situated  that  the  element 
bisects  the  angle  between  the  acceleration  vector  for  the  given 
element  and  the  tangent  to  the  circle  at  the  given  point. 

Property  4.  That  curve  corresponding  to  a  given  lineal  ele- 
ment, which  according  to  Property  2  has  third  order  contact  with 
its  circle  of  curvature  has  a  radius  of  curvature  equal  to  three 
times  the  ratio  of  the  tangential  component  of  the  force  to  the 
normal  component  of  the  space  derivative  of  the  force. 

II.  In  the  second  part  it  will  be  shown  that  the  four  proper- 
ties described  are  sufficient  to  determine  a  set  of  oo 3  curves  of 
constant  pressure  in  a  field  of  force. 

III.  By  varying  the  parameter  c  we  get  &> l  sets  of  curves  Sc 
and  therefore  oo 4  as  the  total  number  of  curves  of  constant 
pressure  in  a  given  plane  field.  The  following  properties  of  the 
quadruply  infinite  system  are  found  to  be  sufficient  to  completely 
characterize  it. 

Property  1.  Those  curves  of  constant  pressure  which  have 
a  given  curvatiire  element  have  osculating  conies  whose  centers 
lie  on  a  conic  tangent  to  the  given  element. 

Property  2.  The  oo3  circles  of  the  plane  are  curves  of  con- 
stant pressure  and  the  pressure  for  these  curves  is  infinite. 

Property  3.  The  radius  of  curvature  for  the  conic  of  Prop- 
erty 1  is  one-tenth  the  difference  of  the  radius  of  curvature  for 
those  curves  having  third  order  contact  with  their  circles  of 
curvature  minus  the  radius  of  curvature  of  the  given  curvature 
element. 

Property  4.  As  the  curvature  of  a  given  element  varies, 
leaving  the  direction  and  point  fixed,  the  second  center  of  curvature 
of  the  conic  of  Property  1  describes  a  cubic  curve  which  passes 
through  the  given  point  once  and  has  an  asymptote  parallel  to  the 
fixed  direction  with  a  double  point  on  the  asymptote  at  infinity. 

Property  5.  The  asymptote  of  the  cubic  of  Property  4  inter- 
sects the  normal  of  the  lineal  element  to  which  the  cubic  belongs 


introduction!  ",-    •»'  :  ■-:■::  ;vii 

at  a  point  whose  distance  from  the  fixed  point  of  the  element  is 
three-tenths  the  ratio  of  the  tangential  component  of  the  force  to 
the  normal  component  of  the  space  derivative  of  the  force. 

Property  6.  The  tangent  of  the  angle  which  the  cubic  of 
Property  4  makes  with  the  normal  to  the  fixed  element  is  one-tenth 
the  sum  of  three  ratios.  The  first  ratio  is  negative  three  times  the 
product  of  the  normal  component  of  the  second  space  derivative  of 
the  force,  as  specialized  for  those  curves  which  have  third  order 
contact  with  their  circles  of  curvature,  multiplied  by  the  tangential 
component  of  the  force  divided  by  the  square  of  the  normal 
component  of  the  first  space  derivative  of  the  force.  The  second 
ratio  is  four  times  the  tangent  of  the  angle  which  the  first  space 
derivative  of  the  force  makes  with  the  normal.  The  third  is  the 
negative  tangent  of  the  angle  which  the  fixed  element  makes  with 
the  force  vector. 

Property  7 .  Of  the  oo l  curves  of  constant  pressure  having 
a  given  curvature  element,  two  have  contact  of  the  fourth  order 
with  their  osculating  parabolas.  The  locus  of  the  foci  of  these 
hyperosculating  parabolas  as  the  curvature  varies,  leaving  the 
lineal  element  fixed,  is  a  quintic  curve  having  a  triple  point  at  the 
origin  and  having  the  element  as  double  tangent.  The  third 
branch  of  the  quintic  at  the  origin  is  so  placed  that  its  tangent  at 
that  point  makes  with  the  line,  of  force  an  angle  bisected  by  the 
element. 

Property  8.  The  quintic  of  Property  7  intersects  the  fixed 
element  tangent  at  a  point  whose  distance  from  the  point  of  the 
element  is  five-halves  the  ratio  of  the  normal  component  of  the 
space  derivative  of  the  force  to  the  normal  component  of  the 
second  space  derivative  of  the  force  for  those  curves  of  constant 
pressure  which  are  hyperosculated  by  the  particular  parabola 
whose  focus  is  the  point  of  intersection  of  the  quintic  and  the  ele- 
ment tangent. 

IV.  Property  1  is  characteristic  of  any  set  of  oo  <  curves 
represented  by  a  certain  general  fourth  order  differential  equation 
and  the  other  seven  properties  stated  above  are  shown  to  be  suf- 
ficient to  specialize  the  equation  and  identify  a  set  of  oo 4  curves 
as  curves  of  constant  pressure. 

The  writer  takes  this  opportunity  to  gratefully  acknowledge 
her  indebtedness  to  Professor  Kasner  for  his  helpful  criticisms 
and  suggestions. 


CHAPTER   I 

Properties  of  the  Set  Sc  of  °o3  Curves  of  Constant 
Pressure  Corresponding  to  One  Value  of  the  Parameter  c. 

1.  The  differential  equation  of  Sc. 

Given  a  plane  field  of  force  in  which  the  general  equations 
of  motion  for  a  particle  moving  freely  are 

(i)  mdi^=<p  (x*y)' m  w  =* <yX' y^' 

Let  a  particle,  whose  mass  may  without  loss  of  generality  be 
assumed  to  be  unity,  be  projected  at  any  time  t  =  t0  from  any  posi- 
tion x  =  x0,  y  =  y0  and  with  any  velocity  defined  by  the  direction 
y'  =  y'0  and  the  speed  v  =  v0.  It  is  required  to  find  a  smooth  curve 
such  that  the  pressure  of  the  particle  against  the  curve  shall 
remain  constant. 

Since  the  curve  is  smooth  the  tangential  component  of  the 
force  vector 

(2)  T  =  v  vg  gives  the  acceleration  at  of  the  particle  along  the  curve. 

The  acceleration  along  the  normal  involves  the  constant  pressure 
P  =  c  and  is  given  by  the  formula 

v2 

(3)  an  =  —  =  N+P  where  N  is  the  normal  component  of  the  force 

vector.  It  is  evident  that  for  a  given  value  of  the  parameter 
P  =  c  the  curve  is  uniquely  determined.  By  varying  the  arbitrary 
constants  x0,  y0,  y'„,  v0  we  obtain  in  all,  since  each  curve  could  be 
described  from  any  one  of  its  points,  oo 3  curves  of  constant  pres- 
sure in  the  plane  corresponding  to  one  value  of  c.  Taking  into 
account  the  various  values  of  c  we  find  oo  of  these  triply  infinite 
systems  of  curves  Sc.  The  system  of  trajectories  forms  one  of  the 
triply  infinite  systems  corresponding  to  c  —  o. 

...  v2 

Eliminating  v  from  the  equations  —  =  N+c  and  v  vs=T  gives 

T 

a  differential  equation  in  intrinsic  form  of  the  system  Sc.  The 
equation  is 

(4)  (N+c)rs  =  2T-rNs. 

To  find  the  equivalent  equation  in  Cartesian  coordinates  the  fol- 
lowing substitutions  are  necessary : 

(1) 


••■•.■■■ 

■    ■  -    • .  •    .  •  . 
..."     ■   •     • 


CURVES    OF    CONSTANT    PRESSURE. 


(5)  N=+^L,     T  =  ^M 

_[*«+(*,-*>,)  y'-<pyy'2}  (l+y")-y"  (y+yV) 

(7)  r=  (i+^!E)ra_3/y^-y"(i+/2) 


r  y"2 


The  result  is 


(8)  y„,  =  y"  (ts-y'vs)  ^i+y'2)  3  y"2  (c  y - ^  V 1  +/*) 


cVi+y2+^_y^  V1+/2  (c  Vi+ya+^-yV) 

This  is  therefore  the  differential  equation  of  Sc. 

2.  Osculating  parabolas. 
Equation  (8)  has  the  form 

(9)  y'"  =  Gy"+Hy"K 

It  has  been  proved  (Kasner,  Trajectories  of  Dynamics,  Transac- 
tions of  Amer.  Math.  Soc.  1906)  that  every  system  of  curves 
represented  by  an  equation  of  this  form  has  Property  1  of  tra- 
jectories.     We  may  therefore  state  the  following 

Theorem  I.  If  in  any  field  of  force  a  particle  is  projected  from 
a  given  point  in  a  given  direction,  to  each  initial  speed  there 
corresponds  a  definite  curve  along  which  the  pressure  of  the 
particle  against  the  curve  remains  constant.  The  00 l  curves 
of  constant  pressure  obtained  by  varying  the  initial  speed  are  so 
situated  that  the  locus  of  the  foci  of  the  osculating  parabolas 
constructed  at  the  given  point  is  a  circle  passing  through  the  given 
point. 

This  theorem  expresses  Property  1  for  the  set  of  00 3  curves 
of  constant  pressure  Sc. 

3.  Hyperosculating  circles. 

Of  the  00 '  curves  of  constant  pressure  having  the  given  lineal 
element  (x,  y,  y')  it  can  be  shown  that  one  has  contact  of  the  third 
order  with  its  circle  of  curvature. 

The  curve  and  its  circle  of  curvature  must  have  the  same 
values  of  y',  y" ',  y'"  at  the  chosen  point  in  order  to  have  contact 
of  the  third  order.     The  differential  equation  of  all  circles  is 

3  y'y'n 

:  i+y 


(10)  y"> 


CONVERSION    OF    PROPERTIES.  3 

Placing  this  value  of  y'"  equal  to  that  in  equation  (8)  for  Sc  and 

simplifying,  we  have  y"  =  ~  ?   ,     7i?        ',  unless  y"  =  o,  which 

3  {<p-\-y\f/) 

leads  to  straight  lines  only  and  is  therefore  excluded  from  the 
present  discussion.  This  shows  that  for  every  value  of  x,  y,  y' 
there  is  one  and  only  one  value  of  y"  under  the  required  con- 
ditions. Therefore  there  is  one  curve  of  Sr  having  third  order 
contact  with  its  circle  of  curvature. 

Moreover,  this  value  of  y"  is  independent  of  c  and  in  fact  is 
exactly  the  value  obtained  in  the  corresponding  discussion  for 
trajectories  when  the  independent  variable  of  the  derivatives 
has  been  changed. 

The  coordinates  of  the  center  of  curvature  for  any  curve  are 

-/ (1+y2)  v_i+/2 


y"       '         /'  ' 

Since  y"  is  determined  by  x,  y,  y'  for  those  circles  having  third 
order  contact  and  is  the  same  for  all  systems  Sc,  it  is  evident  that 
the  center  of  the  hyperosculating  circle  corresponding  to  a  given 
element  is  the  same  for  all  systems  Sc. 

When  y'  varies  the  locus  of  the  centers  of  hyperosculating 
circles  for  Sc  will  be  the  same  as  that  found  for  trajectories,  a 
conic  passing  through  the  given  point  in  the  direction  of  the  force 
in  the  field  (<p,  yp). 

Theorem  II.  In  each  direction  through  a  given  point  there 
passes  one  curve  of  constant  pressure  which  has  third  order  con- 
tact with  its  circle  of  curvature.  The  locus  of  the  centers  of  the  °°  l 
hyperosculating  circles  obtained  by  varying  the  initial  direction, 
is  a  conic  passing  through  the  given  point  in  the  direction  of  the 
force  acting  at  that  point. 

Theorem  II  expresses  Property  2  for  5C. 

4.  The  focal  circle. 

The  equation  of  the  focal  circle  corresponding  by  Theorem  I 
to  a  given  lineal  element  for  the  general  system  (9)  is 

(11)  2G(a2+02) 

+[3  jyi_l)_y  (y*+i)H\a 
+[(y'*+l)H-6y']p  =  o, 

where  a  and  /3  are  coordinates  of  the  focus  of  the  variable  osculating 
parabola  referred  to  the  fixed  point  (x,  y)  as  origin.  Since  in  the 
case  of  curves  of  constant  pressure 


4  CURVES  OF  CONSTANT  PRESSURE. 


and 


//  = 


Vi+y2  (*  Vi+ya+^-y^)' 

the  equation  of  the  focal  circle  becomes 


(13)  2  (*.-/<?.)  Vi+yt  («*+£=) 


+3  [(t'2-1)JH-2  yV-c  Vl+/2]a 

+  3  [-c/  Vl+/2+<p  ty*-  l)-2  /*]  ^  =  o. 

Differentiating  equation  (13)  with  respect  to  a  we  find  that  the 
direction  of  the  circle  at  the  point  a  =  o,  /3  =  o  is  given  by  the  slope 

c  Vl+/2-(/2-l)  y/,-2  y'tp 


■cy'  Vl+ya+(y2-l)  <p-2y') 


The  tangent  of  the  angle  which  this  direction  makes  with  the 
chosen  element  is  then  found  to  be 


If  equation  (13)  is  simplified  by  rotating  the  axes  of  coordinates 
so  that  the  X-axis  coincides  with  the  tangent  at  the  fixed  point, 
thus  making  y'  =  o,  we  have  instead  of  (13) 

2  >£s  (a2+02)-3  (Jt+c)  ffl-3  <p(3  =  o 

and  the  tangent  of  the  angle  between  the  focal  circle  and  the 

element  becomes  the  slope  — — . 

<P 

In  the  case  of  trajectories,  where  c  =  o,  this  slope  reduces  to 

\L        .      .  .  . 

— ,  which  is  known  to  be  the  slope  of  the  focal  circle  corresponding 

to  the  system  of  trajectories  which  pass  through  the  given  point 
in  the  given  direction. 

Returning  to  the  components  of  acceleration  of  the  particle 
moving  along  the  curve  of  constant  pressure,  ax=T,  an  =  N+c, 
the  corresponding  components  along  the  axes  are 

ax=T  cos  6—  (N+c)  sin  0 
au=T  sin  0  +  (AT+c)  cos  0 


CONVERSION    OF    PROPERTIES. 

where  d  =  tan'ly'.     Hence  the  slope  of  the  acceleration  vector 


ay  _     xP  Vl+/3+c 
ax      <p  Vi-j-y2_6y  ' 

Transforming  this  expression  by  rotation  so  that  y'  =  o  it  becomes 
- — .  Since  the  slope  of  the  tangent  to  the  focal  circle  has  been 
shown  to  be ■ —  we  have  proved 

Theorem  III.  The  circle  that  corresponds  according  to 
Theorem  I  to  a  given  lineal  element  is  so  situated  that  the  element 
bisects  the  angle  between  the  acceleration  vector  for  the  given 
element  and  the  tangent  to  the  circle  at  the  given  point. 

This  constitutes  Property  3  for  Sc. 

5.  The  correspondence  between  the  direction  of  the  accelera- 
tion vector  and  that  of  its  corresponding  element. 

We  have  seen  that  for  every  given  element  there  is  a  cor- 
responding focal  circle  whose  direction  at  the  given  point  is  repre- 
sented by  the  slope 


_ c  ^i+y/2-(/2-i)  *- 2-yV 
-cy'  Vl+y2+(/2-l)  <p-2  y'+ 

Also  for  every  element  there  is  a  corresponding  acceleration  vector 
for  the  particle  moving  along  the  curve  of  constant  pressure  which 
has  the  given  element.      The  slope  of  the  acceleration  vector  is 

j,  Vi+ys-fc 

— ,  .      There  are  thus  two  pencils  of  lines  through  the 

given  point  besides  the  pencil  formed  by  the  tangent  of  the  given 
element  as  it  revolves  about  the  fixed  point.  Moreover,  the 
pencils  are  so  related  that  the  tangent  of  the  given  element  always 
bisects  the  angle  formed  by  the  lines  corresponding  to  it  in  the 
other  two  pencils.  A  study  of  these  pencils  furnishes  an  inter- 
esting description  of  them. 

When  the  pencil  of  tangents  represented  by  the  slope  y'  is 
moved  to  a  separate  point  and  the  locus  of  its  points  of  intersection 
with  corresponding  rays  in  the  pencil  of  acceleration  vectors  is 
traced,  the  locus  proves  to  be  a  quartic  curve  of  a  distinct  character. 
Because  this  curve  provides  a  means  of  describing  the  pencil'  of 
acceleration  vectors  in  a  purely  geometric  way  which  is  useful  in 
the  converse  problem  to  be  considered  later,  its  generation  will 
be  given  somewhat  in  detail. 


6  CURVES  OF  CONSTANT  PRESSURE. 

A  simple  form  of  the  equation  of  the  locus  is  found  by  taking 
the  first  chosen  point  (x,  y)  as  origin  and  placing  the  center  of 
the  pencil  of  element  tangents  at  a  point  whose  coordinates  are 
a  <p  and  a  \p  with  reference  to  the  chosen  point  as  origin,  i.  e.,  at 
any  point  on  the  line  of  force.     The  pencil  of  lines  at  the  origin 

y      \l/  Vi -|_3,/2_j_c 

is  then  represented  by  -  = ,    ■        ,    while   the    pencil  at 

.    y  —  a.  ib  .  .  .      . 

{a  <p,  a  \f/)  is =y  ,  corresponding  lines  being  those  which  m- 

%—a  <p 

volve  the  same  value  of  y' . 

Eliminating  y'  from  these  equations  we  have  the  equation  of 

the  locus  of  intersections  of  corresponding  lines.     It  is  found  to  be 

(14)  (v?  y-yf,  x)2  [(x-a  <p)2  +  (y-a  r/02] 
=  c2  [x  (x  —  a  <p)-\-y  (y  —  a  i^)]2. 

The  conclusions  reached  above  may  be  stated  as 

Theorem  IV.  The  correspondence  between  the  elements  at 
a  given  point  and  the  acceleration  vectors  at  that  point  is  such 
that  when  the  first  pencil  of  rays  is  moved  to  a  point  on  the  line 
of  force  the  locus  of  points  of  intersection  of  corresponding  rays  is 
a  quartic  curve. 

6.   Distinguishing  characteristics  of  the  quartic  in  Theorem  IV. 

The  equation  of  the  quartic  of  Theorem  IV  can  be  somewhat 
simplified  by  choosing  the  center  of  the  pencil  of  element  tangents 
at  the  point  (<p,  \p)  thus  making  the  distance  between  the  centers  of 
the  pencils  represent  the  magnitude  of  the  force  vector  "v<p2-j-^2. 
The  equation  of  the  quartic  is  then 

(15)  {.<py-txY[{x-vY+(y-W\ 

=  c2  [x  (%—ip)+y  {y-i)Y- 

When  this  equation  is  differentiated  and  the  slope  of  the  curve 

/  dv\ 

found  at  the  point  (<p,  \p)  we  get  I  <p-\-yp  -f- )2  =  o. 

Therefore  the  curve  has  a  double  tangent  at  (tp,  \p)  and  the  tangent 
is  perpendicular  to  the  line  of  force. 

In  a  similar  way,  there  are  found  to  be  two  distinct  tangents 
at- the  origin  and  their  slopes  are 


dx  '  ^>4+<pV2-cV2 


CONVERSION    OF    PROPERTIES.  7 

The  tangents  of  the  angles  which  these  lines  make  with  the  line 

of  force  -  =  -  are  then  found  to  be ,  i.  e.,  they  are  pro- 

x     <p  ±    yJ^-\-^ 

portional  to  the  positive  and  negative  reciprocals  of  the  length  of 
the  force  vector,  and  the  constant  of  proportionality  is  the  param- 
eter of  the  system  of  curves  of  constant  pressure.  It  is  evident 
also  that  the  'line  of  force  bisects  the  angle  between  the  two 
branches  of  the  quartic.     The  facts  given  above  form 

Theorem  V.  The  quartic  curve  of  Theorem  IV  has  a  double 
tangent  perpendicular  to  the  line  of  force  at  the  center  of  the 
pencil  of  element  tangents  and  two  distinct  branches  which  make 
equal  angles  with  the  line  of  force  at  the  center  of  the  pencil  of 
acceleration  vectors.  The  tangents  of  the  equal  angles  just  men- 
tioned are  proportional  to  the  positive  and  negative  reciprocals  of 
the  length  of  the  force  vector,  the  constant  of  proportionality 
being  the  parameter  of  the  system  of  curves  of  constant  pressure. 

Rotate  the  axes  and  translate  them  so  as  to  place  the  tac-node 
of  the  quartic  at  the  origin  with  the  x-axis  as  the  double  tangent 

and  the  crunode  at  the  point  (o,  ^<p2-\-$2).  The  y-axis  then  co- 
incides with  the  line  of  force  and  the  equation  has  the  form 

x2  (<p2+yfr2)  (x2-\-y2)=c2  {x2+y2-y  Vp*+f*)». 

It  is  apparent  that  the  curve  is  now  symmetrical  with  respect  to 
the  j-axis  which  is  the  line  of  force.  By  differentiating  it  is  found 
that  at  the  origin 


dx2  " 


L  v+*2  d"  c\ 


The  curvature  of  the  two  branches  is  thus  seen  to  be  the  same  as 
the  curvature  at  the  vertex  of  the  two  parabolas 


y  = 


,    l        +  -1  x2  and  y  =  [    .    l         -  -1  x2 


respectively.      Since  the  constant  coefficient  in  these  equations 
gives  the  reciprocal  of  the  latus  rectum  of  the  parabola  the  latus 

rectum  of  the  first  is  — - —  and  of  the  second 


c+  V^+^2  c_  V^2+^2- 

When  the  pencil  of  element  tangents  is  at  any  other  point  on 
the  line  of  force  (a<p,  a\p)  the  directions  of  the  tangents  mentioned 
in  Theorem  V  remain  unchanged  but  the  curvature  of  the  two 


8  CURVES  OF  CONSTANT  PRESSURE. 

branches  at  the  tac-node  vary  with  a,  for  -j—„  =  -      — -. =*=  -    • 

dx2      a  [    V^2_)_^2      c  J 

The  length  of  the  force  vector  V^>2-}-^2  is  thus  identified  as  that 
distance  between  the  centers  of  the  generating  pencils  of  rays 
which  produces  the  simplest  expression  for  the  curvature  of  the 
branches  of  the  quartic  at  the  tac-node. 

Theorem  VI.  The  quartic  of  Theorem  IV  is  symmetrical 
with  respect  to  the  line  of  force.  When  the  pencil  of  element 
tangents  is  on  the  line  of  force  and  at  a  distance  from  the  fixed 
point  of  the  chosen  element  equal  to  the  length  of  the  force  vector 
the  two  branches  of  the  quartic  at  the  tac-node  have  the  same 
curvature  as  that  at  the  vertices  of  two  parabolas :  the  latus  rectum 
of  one  parabola  is  equal  to  the  product  of  the  parameter  of  the 
system  of  curves  of  constant  pressure  multiplied  by  the  length  of 
the  force  vector  divided  by  the  sum  of  the  parameter  and  the 
length  of  the  vector,  while  the  latus  rectum  of  the  other  parabola 
is  the  product  of  the  same  parameter  and  length  divided  by  their 
difference. 

7.  Geometric  construction  of  the  force  vector. 
Properties   2   and   3   furnish   sufficient   information   for  the 

geometric  construction  of  the  force  vector.  The  direction  of  it  is 
given  by  the  direction  of  the  conic  which  according  to  Theorem  II 
is  the  locus  of  the  centers  of  the  hyperosculating  circles  belonging 
to  the  curves  of  constant  pressure  passing  throtigh  a  given  point. 
The  magnitude  of  the  force  vector  is  represented  by  that  distance 
between  the  centers  of  the  pencils  of  rays  generating  the  quartic 
curve  of  Property  3  which  produces  the  simplest  expression  for 
the  curvature  of  the  branches  of  the  quartic  at  the  tac-node. 

8.  The  curve  which  has  a  hyperosculating  circle. 
Returning  to  Property  2  we  see  that  the  curvature  of  the 

curve  of  constant  pressure  corresponding  to  a  given  lineal  element 
which  has  third  order  contact  with  its  circle  of  curvature  is  char- 
acterized by  the  second  derivative 

.      „  =  (*.-yV.)  (l+y'T  , 
y  3  {ip+y'f) 

Hence  its  radius  of  curvature 

3  (y+yW       3  T 

f*-y<t>8      n 

where  ]v  represents  the  normal  component  of  the  space  derivative 


CONVERSION    OF    PROPERTIES.  9 

3   T 

of  the  force  vector.     The  fact  that  Rc  =  -=-  constitutes  the  fourth 

N 

property  for  Sc  and  may  be  stated  as 

Theorem  VII.  That  curve,  corresponding  to  a  given  element, 
which  according  to  Theorem  II  has  third  order  contact  with  its 
circle  of  curvature  has  a  radius  of  curvature  equal  to  three  times 
the  ratio  of  the  tangential  component  of  the  force  to  the  normal 
component  of  the  space  derivative  of  the  force. 


CHAPTER   II 
The  Converse  Problem  for  5,.. 

1.  Nature  of  the  converse  problem. 

It  will  now  be  shown  that  the  four  properties  found  for  5, 
are  sufficient  to  identify  a  triply  infinite  system  of  curves  as  curves 
of  constant  pressure  in  a  field  of  force.  It  will  appear  that  curves 
having  Property  1  are  represented  by  a  general  differential  equa- 
tion of  the  third  order  having  two  undetermined  coefficients, 
functions  of  x,  y,  y'  and  that  each  of  the  other  properties  specializes 
the  coefficients  until  the  equation  (8)  is  developed. 

2.  Converse  of  Property  1. 

Professor  Kasner  has  proved  that  the  only  triply  infinite 
systems  of  curves  possessing  the  property  that  for  every  lineal 
element  the  corresponding  focal  curve  is  a  circle  passing  through 
the  point  of  the  element  are  those  defined  by  a  differential  equation 
of  the  form 

(9)  y'"  =  G  (x,  y,  f)  y"+H  (x,  y,  y')  y"* 

where  G  and  H  are  arbitrary  functions  of  x,  y,  y'.  This  type 
includes  trajectories  and  all  other  curves  of  constant  pressure 
as  special  cases  since  G  and  H  for  these  curves  have  a  particular 
form. 

3.  Converse  of  Property  2. 

A  restatement  of  Theorem  II  avoiding  the  words  of  dynamics 
is  as  follows :  Of  the  o° l  curves  associated  with  a  given  lineal  ele- 
ment one  has  contact  of  the  third  order  with  its  circle  of  curvature. 
The  centers  of  the  o° x  circles,  obtained  by  varying  the  initial  direc- 
tion, lie  on  a  conic  which  passes  through  the  given  point  in  a  fi^ed 
direction  whose  slope  is  co. 

The  general  equation  of  a  conic  passing  through  the  given 
point  in  the  direction  w  is 

(16)  X  A'2-m  X  Y+v  F2+3  (F-«  A")  =o. 

Here  A'  and  \ '  are  referred  to  the  given  point  as  origin  and  X,  \i,  v  are 
any  functions  of  x  and  y. 

If  this  conic  is  to  be  the  locus  of  centers  of  curvature  it  must  be 

—  y>   (\A-y'2)  1+/2 

satisfied  bv  A  =  —       ..     — ,  Y  = £-  in  which  y  and  y    belong 

•>  y"  y> 

to  the  curves  whose  osculating  circles  have  centers  A'  and  Y. 
Substituting  in  (16)  and  simplifying, 

(  10) 


CONVERSION    OF    PROPERTIES.  1  1 

-(1+/2)  (X /»+/*/+*) 


n 


(1?)  y    "  3U+.J0 

a  relation  which  must  be  satisfied  by  X,  n,  v  when  7'  and  y"  belong 
to  a  set  of  curves  having  Property  2.  The  condition  that  curves 
of  set  (9)  have  third  order  contact  with  their  circles  of  curvature 
gives 

(18)  G+Hy>'  =  \^ 

since  /"  =  ]  1?  y*  for  circles  and   /"  =  G  y"+H  y"2  for  sets 

\-\-y  * 

of  curves  having  Property  1. 

Substituting  (17)  in  (18),  that  is  combining  Properties  1  and  2,- 


(19)  G  = 


3/1  f_    (!+/')  (\/2+m/  +  »01 

+  i+y2JL         3(i+co/)      J 


Replacing  G  by  this  value  in  equation  (9)  we  have  the  equation 
of  curves  with  Properties  1  and  2. 


r-[-B^][-^ 


){\y'--+ny'+v) 


3(1+0;/) 


/'+  H  y' 


4.  Converse  of  Property  3. 

To  avoid  the  assumption  of  the  existence  of  a  field  of  force 
Property  3  may  be  restated  employing  the  facts  of  Theorems 
IV,  V  and  VI.  These  theorems  enable  us  to  describe  the  line 
which  forms  with  the  tangent  to  the  focal  circle  corresponding  to 
a  given  element  the  angle  which  is  bisected  by  the  given  element 
without  using  the  fact  that  that  line  is  the  acceleration  vector. 

Property  3  is  then:  There  exists  for  each  element  tangent 
y'  at  the  point  (x,  y)  a  certain  line  through  the  point  such  that  the 
angle  between  this  line  and  the  tangent  to  the  focal  circle  cor- 
responding to  the  element  (x,  y,  y')  is  bisected  by  the  element. 
The  direction  of  this  line  varies  with  the  direction  of  y'  forming 
a  pencil  of  lines  C\  through  the  point  (x,  y)  such  that  if  the  pencil 
formed  by  /,  C2,  is  moved  to  a  point  on  the  fixed  line  of  slope  o> 
the  points  of  intersection  of  corresponding  rays  lie  on  a  quartic 
curve.  The  quartic  has  a  tac-node  at  C2  with  the  double  tangent 
perpendicular  to  y'  =  o>  and  two  distinct  branches  at  d  forming  an 
angle  bisected  by  y'  =  w.  Moreover,  there  exists  a  vector  F  in 
the  direction  y'  —  w  such  that  the  tangents  of  the  angles  which 
the  branches  of  the  quartic  at  C\  make  with  y'  =  co  are  proportional 
to  the  positive  and  negative  reciprocals  of  the  magnitude  of  the 


12  CURVES  OF  CONSTANT  PRESSURE. 

vector,  the  constant  of  proportionality  being  an  independent 
parameter.  The  quartic  curve  is  symmetrical  with  respect  to  the 
line  y'  =  <j).  When  C2  is  placed  so  that  the  distance  C\  C2  =  [F], 
the  curvatures  of  the  two  branches  at  C2  are  the  same  as  those  at 
the  vertices  of  two  parabolas:  the  latus  rectum  of  one  parabola 
is  the  product  of  the  constant  proportionality  factor  mentioned 
above  and  the  distance  between  the  centers  of  the  pencils  divided 
by  the  sum  of  the  same  constant  and  distance,  while  the  latus 
rectum  of  the  other  parabola  is  the  product  of  the  constant  pro- 
portionality factor  and  the  distance  between  the  centers  of  the 
pencils  divided  by  the  difference  of  the  same  constant  and  distance. 

The  information  thus  expressed  is  sufficient  to  determine  the 
direction  of  the  line  corresponding  to  any  value  of  y'  and  therefore 
of  the  tangent  to  the  focal  circle  corresponding  to  a  given  element. 
A  demonstration  of  the  proof  of  this  statement  follows. 

Let  the  fixed  length  of  the  vector  F  described  above  be  marked 
off  from  the  chosen  point  (x,  y)  in  the  fixed  direction  y'  =  co  and 
let  its  projections  on  any  pair  of  rectangular  axes  through  (x,  y) 
be  <p  and  \f/.  <p  and  \p  are  thus  arbitrary  functions  of  x  and  y  in 
that  the  axes  are  arbitrary  but,  once  the  axes  are  placed,  ip  and  \p 
are  determined  by  the  length  of  F  which  may  now  be  expressed 

V+V'2- 

In  the  dissertation  of  Ruth  Gentry,  Plane  Quartic  Curves, 
1896,  the  general  equation  of  a  quartic  having  a  tac-node  is  given 
in  trilinear  coordinates: 

(yz  —  mx2)  (y  z  —  m'x2)  —2  c x  y2z-\-f  x?jy-\-gx2y2-\-dy3z+hx  y3Jrl  yx 
and  the  tac-node  is  at  (x,  y). 

When  this  is  changed  to  rectangular  coordinates  and  the  terms 
containing  odd  powers  of  x  dropped  to  make  the  curve  symmetrical 
with  respect  to  the  j-axis  it  becomes 

(20)  (y  —  m  x2)  (y  —  m'x2)  =g  x2y2+d  y3-\-l  y4. 

The  coefficients  m,  m' ,  g,  d  and  /  will  now  be  determined  by  means 
of  the  facts  under  Property  3. 

Bv  differentiation  it  is  evident  that  :  -  and  — -.  are  the  latus 

m  m 

rectums  of  parabolas  having  at  the  vertex  the  same  curvature  as 

the  two  branches  of  the  quartic  (20)  at  the  origin.      Hence  for 

the  quartic  of  Property  3, 


c+  V+f-  c-  V^+^ 

m  =  — ,  and  m    =  — ,  . 

c  ^<p2+4<2  c  V^+^ 


CONVERSION    OF    PROPERTIES.  13 

Substituting  x  =  o  in  (23),  y2—dy3  —  lyi  =  o.  y2  =  o  indicates 
the  double  point  at  the  origin  and  ly-  —  dy—l=o,  two  other 
intercepts  on  the  y-axis.  In  order  that  these  two  may  coincide, 
placing  the  second  double  point  on  the  j-axis,  the  roots  of  this 

JO 

equation  must  be  equal,  i.  e. ,  d-  —  4  I  =  o.     Hence  I  =  —  — .     Solving 

2 
the  quadratic  then  y  =  —  -,.     This  value  corresponds  to  the  distance 

between  the  centers  of  the  pencils  in  the  quartic  of  Property  3, 

VJi+ji     Hence  d  =  ~^^2  and  /  =  -  ^L_ 

Differentiating  the  general  quartic  (20)  we  find  that,  at  the 

.      /     2\   dy         1    , 

point  (°'w)'/f:=±5^2wd+2  *»'d+4  g.     Therefore  the  tangents 

to  the  quartic  at  the  point  \o,-%\  make  with  the  v-axis  angles  whose 

tangents  are  ±    ,- j— — — Tmr- •     These  values  correspond  to 

V2  m  d+2  md+4  g 

the  tangents  of  the  angles  which  the  two  branches  of  the  quartic 

in  Property  3  make  with  the  line  joining  the  centers  of  the  pencils. 

Therefore  ± r~r^ — ,  ,  ,  A —  =  + 


^  2  m  d-\-2  w'd-f-4  g      —   >l<p2-\-\f/'y 


Substituting  m— , 

c    V+*2 


and  solving  for  g. 


g  = 


m  = , 

C    V^+^2 

V+*2 

^2  +  ^2_2   C2 


*2+v 

Now,  replacing  hi,  m',  d,  g  and  I  by  their  values  the  general  quartic 

becomes,  simplified, 

*2  (*-++*')  0t--  +  v2)=c2  (x*+y*-y  V^+^«)«. 

By  translation  and  rotation  the  axes  may  be  placed  so  that  the 

point  (p,  o)  becomes  (<p,  \[/)  and  (o,    ^<p2+\p-)  is  the  origin.     The 

equation  of  the  quartic  is  then  (15). 

v  —  ^ 

Now  a  pencil  of  lines  through  (<p,  \p),y'  =  : -,  should  intersect 

x  —  <p 

this  curve  in  points  such  that  the  lines  joining  the  points  to  the 


14  CURVES  OF  CONSTANT  PRESSURE. 

origin  will  be  the  lines  described  in  Property  3,  i.  e.,  the  accelera- 
tion vectors  corresponding  to  the  directions  of  the  elements 
through  the  chosen  point  (%,  y).  Substituting  y  =  \[/-\-y'  {x  —  <s)  in 
the  quartic  equation  (15)  gives  the  factor  (x  —  <p)2  =  o  representing 
the  abscissa  of  the  double  point  at  (<p,  \J/)  and 


(+-y'<p)  (ip  Vl+y/2-c/) 
VT+7"2  ty-y'<p+c  Vi+ya 


or 


Hence 


y =4>-\-y'  (x— <p)  = 


Vi+/2  {yp-y'ip-c  Vi+/2) 

(*-?V)  (*  VT+72+c) 


Vl+y'2  (^-3;V+C  V  1+y,2) 


ty-yV)  (^   Vl+y2-c) 

Vi+y2  (^-tV-c  v  1+3/2) 

The  equation  of  the  pencil  of  lines  passing  through  the  origin  and 
the  points  indicated  by  the  pairs  of  coordinates  just   found    is 

y  _    ±i£  Vl+y2+r 

x     ±^  Vi+y2-cy' 


The  ambiguous  sign  is  accounted  for  by  the  fact  that  Vl-f-y'2 
represents  the  secant  of  the  angle  whose  tangent  is  y'  and  its  sign 
depends  on  the  quadrant  in  which  tani'V  lies.  Corresponding 
to  any  value  of  y'  then  we  have  a  line  through  the  origin  satisfying 
the  description  in  Property  3  of  the  acceleration  vector. 

The  converse  problem  is  now  to  find  those  systems  of  curves 
(9)  in  which  each  focal  circle  makes  with  its  corresponding  element 
an  angle  equal  to  the  angle  which  the  element  makes  with  a  certain 
line  whose  direction  is  determined  by  the  element  and  whose  slope  is 


=  *  Vl+/2+c 
"  p  Vi+/2-c/ 

The  slope  at  the  given  point  of  the  circle  (11)  belonging  to  the 

general  system  (9)  is  ^y  _{(l+y>^  H ■ 

Hence  the  tangent  of  the  angle  which  the  focal  circle  makes  with 

3 


the  element  is  — 


3 /-(!+/»)  H' 


CONVERSION    OF    PROPERTIES.  15 

By  the  condition  imposed  in  Property  3  this  must  equal  the 
tangent  of  the  angle  which  the  element  makes  with  the  line  of 
slope  a. 

■  -3  y'-a- 

'  '3/-(l+/')  H  ~~~'  1+  a  yr 

3 

Solving  for  H  we  find  H  =  —, .     Hence  the  onlv  svstems  of  oo3 

y  —a 

curves  having  Properties  1  and  3  are  those  whose  differential  equa- 

3 

tions  are  of  the  form  V"  =  G  y"  -\ — : v"2,  where  G  is  anv  func- 

y  —<y 


j,  Vi+y2_|_c 
tion  of  x,  y,  y' ,  and  a  = .     The  form  of  (9)  when  a  is 

<p  vl+yt—cy' 
substituted  is 


(21)  y"'=Gy"+       3(cy-^i+y2),- 


v'l+y2  (c  ^l+y'2+\l/-y'<p)' 

Now  replacing  G  by  its  value  from  (19)  including  the  value  of  H 
we  have  the  differential  equation  of  curves  having  the  Properties 
1,  2,  3 

(92)  y">  =    (M'/2+m/  +  *)^    yt, 

c  <l-\-y'2+xP-y'<fi  ' 


3(n/-,pVl+/2)  „2 

V1+/2  (c  vi+ya+^-y^)  y 

5.  Converse  of  Property  4. 

Restating  Property  4  using  the  vectors  mentioned  in  a  purely 
geometrical  way  we  have :  The  radius  of  curvature  for  that  curve 
having  the  lineal  element  (x,  y,  y')  which  according  to  Theorem  II 
is  hyperosculated  by  its  circle  of  curvature  is  three  times  the  ratio 
of  the  projection  of  the  vector  F  described  in  Property  3  upon  the 
tangent  to  the  curve  divided  by  the  projection  of  the  space  de- 
rivative of  that  vector  upon  the  normal.     This  gives 

R   =  3  T  _  3  {<p+y'+) 


(23)  .-.  y 


n 


(i+/2r(*.-yV.) 


3  (<?+/*) 

Now  the  curves  represented  by  (22)  have  third  order  contact  with 
their  circles  of  curvature  provided  (23)  is  true,  i.  e., 


16  CURVES  OF  CONSTANT  PRESSURE. 


y>»  =    3yy 


(1+/2) 


{\y'*+ny'+v)  <py"      3  {cy'-<p  Vi+y«)  /'2 


if  we  substitute  for  /'  the  value  given  by  (23).     Making  the  sub- 


to 


stitution,  replacing  w  bv  -  and  simplifying  the  result  we  find 


.    ,,  ,      ,  ,   ■      Vi+/»(*.-yV.) 

A  V  — |-A*  V  +  "  =  " ~~ "• 

Hence   the  requirement   that   the   curves  of  equation    (9)    have 

Property  4  is  equivalent  to  the  substitution  of : — : — — 

for  the  quadratic  expression  X  y'2+M  y'  -\-v  in  the  equation  (9)  as 
already  modified  by  Properties  1,  2,  3,  that  is  in  equation  (22). 
The  equation  is  then  restored  to  the  special  form  (8). 

6.  The  parameter  in  the  equation. 

It  remains  to  show  that  this  equation  can  represent  no  other 
curves  than  curves  of  constant  pressure  in  the  field  of  force  dp,  \p). 
The  existence  of  the  field  of  force  has  been  established  when  the 
functions  <p  (x,  y)  and  \f/  (x,  y)  were  found  for  every  point  (%,  y) 
in  the  plane.  It  must  now  be  proved  that  the  parameter  c  can  be 
accounted  for  only  by  a  constant  normal  component  of  acceleration 
compelling  the  particle  in  the  field  to  move  along  the  curves  whose 
characteristics  are  being  studied. 

Equation  (8)  may  be  written 


yl„  =  y"  [(*.-/».)  (l+f)+3f  (cy-gV !+/»)] 
Vl+/2  (€  Vi+y2+^-yV) 

Observing  the  form  of  the  denominator  of  the  right  hand  member 
of  the  equation,  it  appears  that  when,  at  a  given  point,  y'  is  such 

that  c  ^l-\-y'2+\f/  —  y'<p  =  o.  it  can  be  proved  that  the  curves  of  (8) 
are  straight  lines.  This  condition  is  equivalent  to  c+N  =  o. 
Excluding  the  discontinuous  case  y'"  =  <»  , 

the  numerator 

y"l(*.-yV.)  U+y'2)+3y"  (c/-*  Vi+/*)]=o. 

Hence  y"  =0 

3  (<:/-«>  Vl+/»)  3TV       • 


CONVERSION    OF    PROPERTIES.  17 

The  first  possibility  makes  y"f  =  o  directly  and  therefore  succeeding 
derivations  are  zero  and  the  curves  are  straight  lines. 

N  f 

If  y   =  —  —  y  (l+y'2)     consider  the  space  derivative  of  the 

normal  component  of  force. 

Since  N  =  —  c,  N,  =  o. 

—    T  .  .  3  7 

But  N,  =  N and  r  for  this  value  of  y"  is  equal  to  -=-. 

r  N 

.  ■ .  N,  =  N  -  y  =  f  N.     Since  Nt  =  o,  N  =  o. 

Consequently  y"  =  o,  y'"  =  o  .  .  . 

Thus  both  possibilities  lead  to  the  conclusion  that  at  every  point 
when  the  slope  is  such  that  N+c  =  o  the  curves  are  straight  lines. 
Since  the  particle  can  move  in  a  straight  line  only  when  it  is  affected 
by  a  tangential  component  of  acceleration  alone,  it  is  evident  that 
c  is  a  normal  component  of  acceleration  which,  when  the  element 
tangent  is  in  this  direction,  cancels  the  normal  component  of  the 
force  (<p,  \f/)  and  renders  the  normal  acceleration  of  the  particle 
zero — that  is,  can  be  nothing  but  a  constant  pressure  between 
the  particle  and  the  smooth  curve. 


CHAPTER   III 

Properties  of  the  Quadruply  Infinite  System  of  All 
Curves  of  Constant  Pressure  in  a  Plane  Field  of  Force. 

1 .  The  intrinsic  equation  for  the  system  of  » 4  curves. 

Since  one  set  of  oo 3  curves  Sc,  corresponding  to  one  value  of 
the  parameter  c,  is  represented  by  the  intrinsic  equation  (4) 
(N-\-c)  rs  =  2  T  —  r  Ns,  if  we  differentiate  this  equation  and  elim- 
inate c  we  shall  have  a  differential  equation  of  the  fourth  order 
representing  the  °o 4  curves  of  constant  pressure  in  the  field  of 
force  (<p,  yp). 

The  space  derivative  of  the  force,  regarded  as  a  vector,  will 
be  used  here.  The  normal  and  tangential  components  of  this 
vector  are 

(24)  n  =  ^s~y,<ps  =  ^*+(^y-p*)  y'-<Pv  y'2 

VI+/2  "  l  +/2 

j  =  <P,+y'iPs  =  <px+(<py+tx)  y'+Tpyy"1 

Vi+y2 "  1+y2 

These  components  are  related  to  the  space  derivatives  of  the 
components  of  force  in  this  way: 

—    T  -    N 

(25)  Ns=N--,T.  =  T+-. 

r  r 

The  space  derivative  of  N  also  will  be  necessary. 

(26)  Ne  =  Nl-\---  where 

r 

Ni  =  ypxx-\-(2  ypxv-Vxx)  /  + 0/^-2  ipxu)  y"2-<Pyyy'3 
=      ypy-2  (<Py+ypx)  y'+ivx-^y)  y'2-<p* 

.   2  ~  1+y2 


Using  (25)  the  equation  (4)  becomes 

(N+c)  rs  =  3  T-Nr. 
Differentiating  with  respect  to  5  and  eliminating  c  we  obtain  the 
intrinsic  equation. 

(27)  (Nr-3  T)  rrs  =  (2  rN-T)  rs2 

+[Ni  r2+(/V2-3  T)  r-3N]r, 
which  represents  the   <»4  curves  of  constant  pressure. 

(  18) 


CONVERSION    OF    PROPERTIES. 


19 


2.  The  equation  of  the  system  in  Cartesian  coordinates. 

In  order  to  investigate  the  characteristics  of  the  system, 
equation  (27)  may  be  translated  into  the  corresponding  Car- 
tesian equation.  The  values  of  N,  r,  rs,  T,  Ni,  N2  and  N  have 
already  been  mentioned  in  (24),  (7),  (5),  (26). 


(28)      r„  = 


y»%  (3y„2 _  2y,y„,)  _  y,yi v( x  +y>2)+2y'">{  1  +/ 2) 


y"3  Vi+/2 

After  making  these  substitutions  and  simplifying,  the  equation 
(27)  becomes 

(29)    yIV  [3  y"  (<p+y'+)- {&+&,- *J  y'-^y'2]  (i+/2)] 

=  y'"2  5  (*+/*) 


+/ 


+9y 


,"4 


y"2  (+-y'<p) 
\+y'2 

-y"[lOy'i+x+(+v-<Px)y'-<Pvy'2} 

++y  (i-4/2)+^  (y2-4)-5  /  (<p„+*z)] 

-(i+y2)i^+(2  +xy-<pxx)  y+^yy-2  <pxy)  y'"- 
-<pyyy'3\ 


<p 


\+V2 


+ 


3y//3  r(5/2-i)i^+(^-^)y-^/2i 

+3y'y"rti'xx+(2txy-<pxx)  y'+(fyy-2  <pIy)  y'2-<Pyyy's} 

To  simplify  the  writing  of  this  equation  let 

(30)  ?!-*.+  (*,-*) /-W^1 

P2  =  ^(l-4/2)4-^x(/2-4)  -5/(^4-^) 

P3  =  ^xx+(2  *„-*«)  /+(*»»- 2  *„)  /2-^v9/3 

D  =  3y/(^+>'V)-(14-/2)P1. 
We  then  have 

(31)  Tiv=5(^+yV)y,2 


D 


/"    r-3(^-y 

z?(i+/8)|_+  (i+/2 


-3  (+-y'<p)  y/j4-(i+y«)  (io  yPi+p2)  y" 

)2Pa 


"4 


+ 


9yy 
D(i+/2) 


20  CURVES  OF  CONSTANT  PRESSURE. 

s  [(s  j^-d  ft+/fti /*• 


+ 


D(l+/2) 


3  y'P3  ..,„ 


It  should  be  noticed  that  Pi,  P2,  P3  are  functions  of  x,  y,  y'  but  D 
involves  also  y" . 

3.  Osculating  conies. 
Equation  (31)  is  of  the  form 

(32)  jr*rAy"'*+B?"+C. 

Professor  Kasner  has  proved  (Bulletin  Amer.  Math.  Soc,  March, 
1907,  vol.  3,  p.  290)  that  the  system  represented  by  such  an  equa- 
tion has  the  characteristic  property  that  the  curves  having  a  given 
curvature  element  (x,  y,  y',  y")  have  osculating  conies  whose 
centers  lie  on  a  conic  tangent  to  the  element. 
We  may  therefore  state 

Theorem  VIII.  Curves  of  constant  pressure  having  a  given 
curvature  element  {x,  y,  y\  y")  have  osculating  conies  whose 
centers  lie  on  a  conic  tangent  to  the  given  element. 

This  will  be  known  as  Property  1  for  the  quadruply  infinite 
set  of  curves  represented  by  equation  (29).  The  quadruply 
infinite  set  of  curves  will  hereafter  be  called  5. 

4.  Circles  as  curves  of  infinite  pressure. 

Considering  the  possibility  that  some  of  the  °°  curves  having 
a  given  curvature  element  might  have  contact  of  the  third  or 
fourth  order  with  their  circles  of  curvature,  those  curves  would 
have  to  agree  with  circles  in  derivations  up  to  the  third  or  fourth 
derivative. 

3  y'v"2  3  y"z  (5  v'2+D 

(33)  For  circles  y'"  =   ,  ,     ,_  and  vIV  =  — ,.  .   '  ,'n • 

1+7"  (1  +T  ) 

3  VV'2 
Substituting  /"  =      \_   ,,    in  (31)  we  find 

(%L.  1V       3  v">  (5  y'g+1)  [3  y"  (y+yV)-Pi  (l+/g)] 

(     }         y  (1+y2)2  [3  y"  {*>+?*) -Pi  d  +  v'2)] 

If  3  y"  (tp-\-y'^)  —  P\   (l+y"2)^o  that  factor  may  be  cancelled, 

3  v"3  (5  V2+ 1 ) 
ylv  =      ,       v  ./,    — -  and  for  every  value  of  v"  all  the  curves  have 

(l+y2)L 
fourth  order  contact  with  their  circles  of  curvature.     In  fact,  the 


CONVERSION    OF    PROPERTIES.  21 

curves  are  then  circles  since  succeeding  derivatives  for  them  must 
be  obtained  from  y';  y",  y'",  ylv. 

Referring  to  the  third  order  problem  of  Sc  and  solving  equa- 
tion (8)  for  c,  we  find 


[P,  y"-3  y  yi-V"  (^-vV)]  Vl+/2 
C  v'"(1+t'-)-3  f  y"2 

3  y'y'"1 
If  y'"  =  .  .     >o  >  as  it  must  if  the  curves  are  circles,  the  denomina- 
\+y- 

tor  of  the  above  fraction  becomes  zero  and  the  numerator, 
y"  [3  y"  (^+/^)-Pi(14\V2)].  It  is  evident  then  that  c=  oo 
when  the  constant  pressure  curves  are  circles  unless  y"  =  o  (when 
the  curves  are  straight  lines))  or  3y"(<p+y'ip)  —  Pi(l+r'2)  =o 
which  is  known  to  be  the  condition  for  third  order  contact  with 
circles  in  every  Sc.  Sx  then  consist  of  co 3  circles  and  these  are 
included  among  the   °°4  curves  that  satisfy  equation  (31). 

Theorem  IX.  The  oo 3  circles  of  the  plane  are  included  among 
the  curves  represented  by  the  given  fourth  order  differential 
equation.  The  pressure  of  the  moving  particle  against  these 
circles  is  infinite.  This  theorem  expresses  Property  2  for  the 
system  5. 

5.  The  hyperosculating  circles. 

If  3  y"  {<p+y'\P)-Px  (1  +/2)  =  o,  or  briefly  D  =  o,  we  have 

(35)  ;;  =3^+7*T 

It  is  known  that  in  every  Sc  the  curve  corresponding  to  a  given 
lineal  element  and  having  the  curvature  represented  by  (35)  has 
third  order  contact  with  its  circle  of  curvature.  Hence  of  the  oo 2 
curves  in  5  corresponding  to  a  given  lineal  element  the  circles  of 
curvature  of  oo  >  curves,  one  from  each  Sc,  coincide.  As  the  direc- 
tion of  the  element  varies  the  center  of  this  hyperosculating  circle 
describes  a  conic  passing  through  the  given  point  in  the  direction 
of  the  line  of  force.  This  is  the  conic  of  Property  2  for  every  Sc. 
It  is  also  known  from  the  third  order  problem  that  the  radius  of 
the  hyperosculating  circle  mentioned  above  corresponding  to  a 
given  lineal  element  is  three  times  the  ratio  of  the  tangential 
component  of  the  force  to  the  normal  component  of  the  space 

3  T 
derivative  of  the  force.     R,  =  -^=^. 

N 


22  CURVES    OF    CONSTANT    PRESSURE. 

6.  The  length  of  the  force  vector. 

In  order  to  find  geometrically  the  length  of  the  force  vector 
in  the  fourth  order  problem  it  is  necessary  to  investigate  the 
relation  of  the  circles  of  Property  2  in  5  to  the  quartic  of  Prop- 
erty 2  in  Sc.  The  acceleration  vectors  when  the  constant  pressure 
is  infinite  fall  along  the  normals.  Consequently  the  points  of 
intersection  of  these  lines  with  the  corresponding  tangents  when 
the  pencil  of  tangents  is  moved  to  the  point  Ci  at  the  extremity 
of  the  force  vector  lie  on  a  circle.  This  circle  is  the  limiting  form 
of  the  quartic  in  Sc  as  c  approaches  infinity  and  its  diameter  is 
the  required  length  of  the  force  vector. 

Theorem  X.  The  °° 1  acceleration  vectors  for  the  circles  of 
Theorem  X  which  pass  through  a  given  point  are  perpendicular 
to  the  tangents  to  the  circles  at  that  point.  When  the  pencil 
of  tangents  is  moved  to  the  point  at  the  extremity  of  the  force 
vector  the  points  of  intersection  of  the  tangents  with  correspond- 
ing acceleration  vectors  form  a  circle  which  is  the  limiting  form  of 
the  quartic  of  Theorems  IV,  V  and  VI,  and  whose  diameter  is 
the  length  of  the  force  vector. 

Having  found  the  length  of  the  force  vector  its  projections 
on  the  tangent  and  normal  for  any  lineal  element  as  well  as  the 
projections  of  the  space  derivative  of  the  force  vector  can  be  found. 

7.  The  radius  of  curvature  of  the  conic  of  Property  1. 

The  equation  of  the  conic  of  Property  1  belonging  to  the 
general  system  (32)  is 

(36)  3  y"3  (3  y"A  -  5)  X-+3  v"2  B  X  (Y-y'X) 

+  C  (Y-y  X)--3  /"  (Y-y  X)=o 

(Kasner,  Bulletin  Amer.  Math.  Soc,  March,  1907.) 
In  the  present  problem 

07)  A  =  5^+y'» 


B=-  1 


D 

-3  OA-vV)  y'n 

+(io/p1+)p2(i+/2)y 

+  (1+/2)2P3 


c  = 


D{\+y'*) 

9^y"4+3  j(5y'2-  1)  Pi+y'Pz]  y"3 


1 P 


£(l+/2)[      +3  y'/Ml +/'2) /'2 
where  Pu  P>>,  Ps,  D  represent  the  values  assigned  them  in  (30). 


CONVERSION    OF    PROPERTIES.  23 

Substituting  (37)  in  (36)  we  have  the  conic  which  is  the  locus 
of  centers  of  osculating  conies  for  curves  having  a  given  curvature 
element  in  5. 

(38)    A'2  [3  tW2+(5-/2)  Piy"-P2y'y"-Psy'  (l  +y'2)] 

+x  y  [-3  a-/?)  y/2+i2  p1yy/+p2y'(i-y2) 

+  P3(l-y'4)] 

+  V2  [3  ^  y"2+  { (5  y'2- 1)  Pi+y'P2]  y"+Pi/  (i+/2)] 

+  X  D  y'  (1+/2)  -YD  ( 1  +y"-)  =0. 

Differentiating  (38)  with  respect  to  X  we  find 

,       ,  10y"P,(l+/2) 

r       y ,  ^ 

The  radius  of  curvature  for  conic  (38)  is  then 


(39)        dV  1+3/2      Vi+y 


10  /'A      1 


_  j_  f3  (y+yv)  vi+72    d+y2)3H    j_  r»   J 

~To[_  Pi  y"      J      10  L        j 

This  is  Property  3  and  is  expressed  by 

Theorem  XL  The  radius  of  curvature  of  the  conic  corre- 
sponding to  a  given  curvature  element  by  Theorem  IX  is  one-tenth 
the  difference  of  the  radius  of  curvature  for  those  curves  having 
third  order  contact  with  their  circles  of  curvature  minus  the  radius 
of  curvature  of  the  given  curvature  element. 

8.  The  locus  of  the  second  center  of  curvature  of  the  conic 
of  Property  1  is  a  certain  cubic. 

The  general  formulas  for  the  second  center  of  curvature  of 
a  given  curve  are 

(Mis              r  -  -4y'(l+;/2)      /"(!+/') 
(4°)  *2  = y, 1 yr3 ; 

(1+y2)  (1-3/2)    ,  yV"(l+y'2)2 

y*  = j, +  —jh • 

For  the  conic  of  Property  1,  Y'=y', 
10  V'P,  (1+/2) 


y= 


D 


„,_ 30  y"Pi[ -3  U-y'v)  y"2+(l+y'2)  (10  y'Px+P2)  y'H 
D*  +(l+y'2)2P3.  J 


24  CURVES  OF  CONSTANT  PRESSURE. 

Substituting  these  derivatives  for  y',  y",  y'"  in  (40), 

[-  10  y'y"  (1+/2)  Pl  Z)  +  3  D  [_3  ty-ftp)  /"] 

(AU    ,_  L  -p2(i+/2)yr+  (i+y)8P«i         •    J 

K*l)   x  ~  ioo  y2  (i+/2)  Px2 


^2: 


fio  y  (i+y)  Pi  £+3  yz?  [-3  (tA-yV)  y2] 

L  +p2  d+y2)  y+  a+y/8)p»i J 


ioo  y2  (i4-/2)  Pi5 


To  find  the  locus  of  the  second  center  of  curvature  of  conic  (38) 
as  y"  varies,  keeping  y'  constant,  y"  must  be  eliminated  from  the 
above  equations  (41).  The  elimination  gives  the  following  result, 
when  the  equation  is  simplified. 

(42)  3  P3  (}'2~y'x2)  [3  (1+/2)  (*+yV)-ro  Pi  (>-2-A2)]2 

4-  Pi  (1+/2)  [3  (1+y2)  (^4-/^-10  Pi  (r2-A2)] 
■  [-  10  Pi  6>2/+*2)+3  P2  (y2-y'x2)} 

-9Pi2(i+y2)  («£-y>)  (^2-y2)=o. 

The  general  character  of  this  curve  is  more  easily  seen  by  writing 
the  equation  in  the  form  when  y'  has  the  special  value  zero. 

(43)  3  P3y2  (10  Pij2-3  <p)2 

+ Pi  (10  P& -Zip)  (10  Pix2 - 3P2y2) 
-9  P^y2  =  o. 

The  curve  is  a  cubic  passing  through  the  origin  once  and  having  an 
asymptote  parallel  to  the  direction  of  the  element  y'  which  passes 
through  a  double  point  at  infinity.  Every  line  parallel  to  the 
element  thus  cuts  the  curve  in  only  one  finite  point.  Property  4 
may  now  be  stated  as 

Theorem  XII.  As  the  curvature  of  a  given  element  varies, 
leaving  the  direction  and  point  fixed,  the  second  center  of  curva- 
ture of  the  conic  of  Theorem  VIII  describes  a  cubic  curve  which 
passes  through  the  fixed  point  once  and  has  an  asymptote  parallel 
to  the  element  with  a  double  point  on  the  asymptote  at  infinity. 

9.  The  intercept  of  the  asymptote  of  the  cubic  on  the  normal. 

Directly  from  the  equation  (42)  we  see  that  the  equation  of  the 
asymptote  is  3(1  +y'-)    (^+/^)-10    Pi    {y'1-y'x1)=Q    and  its 

intercept  on  the  r-axis  is  3  - ^         — Tr±.      The  projection  of 


CONVERSION    OF    PROPERTIES.  25 

.3  (14-/2)      v+y'ip 

this  intercept  on  the  normal  gives  —  - — ^ — .        ,     and  this 

6         10       Pi  V1+/2 

is  the  intercept  of  the  asymptote  on  the  normal  since  the  asymptote 

is  perpendicular  to  the  normal.     The  intercept  may  be  expressed 

3    T 
in  terms  of  known  vectors  — -  v?  and  furnishes  Propertv  5  for  5. 

10  AT  F      - 

Theorem  XIII.  The  intercept  of  the  asymptote  of  the  cubic 
of  Theorem  XII  on  the  normal  to  the  element  to  which  the  cubic 
belongs  is  three-tenths  the  ratio  of  the  tangential  component  of  the 
force  to  the  normal  component  of  the  space  derivative  of  the 
force. 

10.  The  direction  of  the  cubic  at  the  origin. 

The  slope  of  the  cubic  (42)  at  the  origin  is 

[9  p,/(v?+/^)2+ 10^2(^4-/^)4-3  P,P2/(^4-j-V)  1 

,     L -3  Pry  (iA-.-vV) J 

y*  "  \9  p3(^4-Jv)2-io  PiVO+yW+3  PiP2(<p+y'+)  1 

L  -3  PS  (+-y'<p)  J 

whence  the  tangent  of  the  angle  which  this  direction  makes  with 
the  normal  to  the  chosen  element  is 

<«>  -3[3P,(^)+^-^^} 

10  Pi2 

In  order  to  obtain  a  geometric  interpretation  for  this  expression 
it  is  necessary  to  write  down  the  components  of  the  second  space 
derivative  of  the  force  vector. 

The  normal  component  is  found  to  be 

(14-/2F 

(i+/»r 

For  those  curves  which  have  third  order  contact  with  their  circles 

*  +  t.        r/    Pi(l+v'2) 

of  curvature,  when  V  =  „  /    ,   ' ,  ' , 

3  (<P+y  W 

ft=ft        3P,(y+yV)+P#l,-y(yl>+^)+3rfV,] 

3  (*+•*)  (14-/^ 

Now  P2  =  *„(l-4  /2)-f  Vx(/2-4)~S  /{**+**) 


26 


CURVES  OF  CONSTANT  PRESSURE. 

.-.  3  (*>+/*)  P3+PiP2  =  Nc  3  (?+/*)  (l+y2)': 

=ATe3  7(l+y2)2-4AT  •    T  (1+/2)2 
(44)  becomes 


_3_ 

10 


To 


3  JVer-4  iV  T+N2  ^^    /} 


N2 


3NCT      AT 

W       iv 


l 


This  is  Property  6  and  may  be  expressed  in  words  as 

Theorem  XIV.  The  tangent  of  the  angle  which  the  cubic  of 
Theorem  XII  makes  with  the  normal  to  the  element  to  which  the 
cubic  corresponds  is  three-tenths  the  sum  of  three  ratios.  The  first 
ratio  is  negative  three  times  the  product  of  the  normal  component 
of  the  second  space  derivative  of  the  force,  as  specialized  for  those 
curves  which  have  third  order  contact  with  their  circles  of  curva- 
ture, multiplied  by  the  tangential  component  of  the  force  divided 
by  the  square  of  the  normal  component  of  the  first  space  derivative 
of  the  force.  The  second  ratio  is  four  times  the  tangent  of  the 
angle  which  the  first  space  derivative  of  the  force  makes  with  the 
normal.  The  third  is  the  negative  tangent  of  the  angle  which  the 
fixed  element  makes  with  the  force  vector. 


(45) 


1 1 .  Hyperosculating  parabolas. 

The  differential  equation  for  all  parabolas  in  a  plane  is 
5  y'"2 


y'2  = 


3/ 


Those  curves  of  S  which  have  contact  of  the 


fourth  order  with  parabolas  must  have  the  same  value  of  yiv. 
Substituting  (45)  in  (31)  the  equation  is  seen  to  be  quadratic  in 
y'" .  Hence  of  the  oo1  curves  of  (31)  having  a  given  curvature 
element  two  have  contact  of  the  fourth  order  with  their  osculating 
parabolas.  The  locus  of  the  foci  of  these  hyperosculating  parabolas 
as  y"  varies  will  now  be  found.  The  coordinates  of  the  focus  of 
a  parabola  in  terms  of  derivatives  are 


(46) 


z  a 


2(3- 


CONVERSION    OF    PROPERTIES.  27 

-3  y»[y'"(y'«—\)+2  y'(3  y'2-//")] 

y'"2+(3  y"*-y'y'"y 

-3  y"[(3  y'"l-y'y"')  (/2-l)-2  //"] 


}'"'24-(3  y"*-y'y'"y 

Solving  these  equations  for  y"  and  y'" 
(14-/2)  (£-«/) 


(47)  /'  = 


2  (a24-/32) 


„,  =  3  (1+/2)  (/3-a  /)  [a+/3  y'  +  y'  (/3-a  y')) 
y  4  (a24-/32)2 

When  /',  7"'  and  yIV  from  (47)  and  (45)  are  substituted  in  (31) 
the  locus  of  the  focus  (a,  /3)  as  y['  varies  will  be  found,  a  and  /3 
being  referred  to  the  chosen  point  (x,  y)  as  origin.  The  resulting 
equation  simplified  is 

(48)     4  P3  (a24-/32)2(«  +  /3  y')  -  10  Pi(a24-/32)  (a+/3  y')2 

4-2  Pifas+/3«)  (/3-a  y')2  +  2  P2(a2  +  /32)  (a+0  V)  (/3-a/) 
-3  ty-y<p)  (/3-a  y)2 (a +/3  /) 
-3  (*>+/$  (0-a  y'y  =  o 

representing  a  curve  of  the  fifth  degree  having  a  triple  point  at  the 
origin  and  the  element  as  a  double  tangent. 

Theorem  XV.  Of  the  °° '  curves  of  5  having  a  given  curvature 
element  two  have  fourth  order  contact  with  their  osculating  para- 
bolas. The  locus  of  the  foci  of  the  hyperosculating  parabolas  as 
the  curvature  varies,  leaving  the  lineal  element  fixed,  is  a  quintic 
curve  having  a  triple  point  at  the  origin  and  having  the  element 
tangent  as  a  double  tangent. 

The  slope  of  the  third  branch  at  the  origin  is  found  to  be 

—     ,     . ,  ,     .   , — ^-;-\  and  the  angle  which  this  line  makes  with  the 

v+y't+y'W-yv) 

element  is  the  same  as  that  which  the  element  makes  with  the 
line  of  force. 

Theorem  XVI.  The  single  branch  of  the  quintic  of  Theorem 
XV  at  the  fixed  point  is  so  situated  that  its  tangent  at  the  fixed 
point  makes  with  the  line  of  force  an  angle  bisected  by  the  element. 

Theorems  XV  and  XVI  express  Property  7. 


28  CURVES  OF  CONSTANT  PRESSURE. 

12.  The  point  of  intersection  of  the  quintic  with  the  chosen 
element. 

Let  jS  —  ay'  =  o.     We  find  then 

5  P 
(49)  a  =  — p1  and 

P       2P3 

as  the  coordinates  of  the  only  point  of  intersection  of  the  element 
tangent  with  the  quintic  (48)  in  addition  to  the  points  at  the  origin. 
The  distance  from  the  origin  to  this  single  point  Q  of  intersection  is 


5  Px  Vl+V2  =  5iV_ 
2  P3  "  2  Ni  ' 

In  seeking  the  geometric  explanation  of  this  quantity  it  is  necessary 
to  observe  that  W.-ff-^'"^^*^1. 

(1+?      )  ' 

Now  since  the  quintic  of  Theorem  XVI  is  the  locus  of  foci  of  oscu- 
lating parabolas  the  point  Q  is  the  focus  of  a  particular  parabola. 
Using  again  the  general  coordinates  of  the  focus  (46)  and  giving 
them  the  special  values  (49)  it  is  found  that  the  only  curvature 
possible  in  this  case  is  given  by  y"  —  o.     Hence  we  may  say  that 

N\  =  Ntt  where  Na  indicates  the  normal  component  of  the  second 
space  derivative  of  the  force  for  those  curves  which  are  hyperos- 
culated  by  the  particular  parabola  whose  focus  is  Q.  We  now 
state  as  Property  8  the  facts  just  proved. 

Theorem  XVII.  The  quintic  corresponding  to  a  given  lineal 
element  by  Theorem  XV  intersects  the  element  tangent  at  one 
point  besides  the  point  of  the  element.  The  distance  of  this 
point  from  the  point  of  the  element  is  five-halves  the  ratio  of 
the  normal  component  of  the  first  space  derivative  of  the  force 
to  the  normal  component  of  the  second  space  derivative  of  the 
force  for  those  curves  of  constant  pressure  which  are  hyperosculated 
by  the  particular  parabola  whose  focus  is  thus  defined  by  the  inter- 
section of  the  quintic  and  the  element  tangent. 


CHAPTER   IV 

The  Converse  Problem  for  5. 

1.  Converse  of  Property  1. 

Professor  Kasner  has  proved  that  Property  1  is  characteristic 
of  all  sets  of  °o 4  curves  whose  equations  are  of  the  form 
(32)  yiy=A  y'"-+B  y'"+C,  where  A,  B  and  C  are  arbitrary 
functions  of  x,  y,  y',  y" .  It  will  now  be  shown  that  the  other 
properties  already  stated  are  sufficient  to  specialize  the  equation 
(32)  until  it  takes  the  form  (31)  which  is  known  to  represent  all 
the  curves  of  constant  pressure  in  a  plane  field  of  force. 

2.  Converse  of  Property  2. 

Since  the  circles  of  a  plane  are  included  in  the  system  (32)  the 
differential  equation  of  the  circles  must  satisfy  the  equation  (32). 

Substituting  V"  =  ^C,  -vIV  =V'1(^2C+1)  in  (32)  we  have 

1+y-  (1+/*) 

(50)  3/'3(5/2+l)=VyM4+3/y/2(l+/2)  B+{\+y'"YC, 

a  relation  among  A,  B  and  C  in  consequence  of  Property  2.  Solv- 
ing for  C  and  substituting  the  resulting  value  in  (32)  we  have  the 
equation  of  all  curves  possessing  Properties  1  and  2. 

(51)  f*=A  y'"2+ B  y'" 

.$  y"z{5  y'2+\)-9  y'ytA-Sy'y"*  (1+/2)  B 

(1+/2)2 

C  will  still  be  written  instead  of  the  long  final  term  but  it  is  under- 
stood that  it  may  at  any  time  be  replaced  by  its  value  in  terms  of 
A  and  B  as  A  and  B  are  modified  by  later  properties. 

3.  Converse  of  Property  3. 

In  Property  3  we  must  make  use  of  all  the  information  implied 
in  the  expression  "radius  of  curvature  of  those  curves  which  have 
third  order  contact  with  their  circles  of  curvature."  In  the 
converse  problem  the  force  vector  and  its  derivatives  are  to  be 
treated  as  purely  geometrical  vectors.     The  force  vector  will  be 

called  F  and  its  first  and  second  space  derivatives  F  and  F  respect- 
ively. The  projections  of  F  on  the  tangent  and  normal  for  any 
given  element  will  be  called  T  and  N  and  the  corresponding  pro- 
jections of  F  and  F  will  be  T,  N,  T  and  N.  Expressed  without 
assuming  the  existence  of  a  field  of  force,  the  information  involved 
in  Property  3  is  then  as  follows : 

(29) 


30  CURVES  OF  CONSTANT  PRESSURE. 

For  each  point  in  the  plane  there  exists  a  vector  F  determined 

in  direction  and  magnitude  by  the  curves  of  5  passing  through  the 

point  which  have  third  order  contact  with  their  circles  of  curvature. 

For  every  lineal  element  at  the  point  there  is  one  circle  of  curvature 

which  has  four  points  in  common  with  each  of  oo1  curves  of  the 

system  5.     As  the  direction  of  the  element  varies  the  center  of  this 

circle  describes  a  conic  which  passes  through  the  fixed  point  in  a 

certain  direction  indicated  hereafter  by  y'  =  <a,  the  direction  of  the 

vector  F.     The  hyperosculating  circle  itself  is  a  member  of  the 

system  5.     The  set  of  all  circles  in  the  plane,  called  5«,  forms  a 

distinct  part  of  5  such  that  when  the  corresponding  tangents  to 

focal  circles  are  considered  according  to  Property  3  of  the  third 

order  problem  a  circle  appears  as  the  limit  of  the  quartic  in 

Property  3.     The  diameter  of  this  limiting  circle  is  the  length  of 

the  vector  F.     The  radius  Rc  of  the  hyperosculating  circle  for  any 

3  T 
element  (x,  y,  y')  is  then  geometrically  expressed  by  — =.     The 

radius  of  curvature  of  the  conic  of  Property  1,  called  briefly 
conic  I,  is  one-tenth  the  difference  of  Rc  minus  the  radius  of 
curvature  for  the  curvature  element  to  which  conic  I  corresponds. 
When  the  vector  F  is  referred  to  any  pair  of  rectangular  axes 
through  the  fixed  point  its  projections  on  the  axes  may  be  called 

>p  and  \p  and  its  direction  w  =  -.     <p  and  \p  are  thus  functions  of  x  and 

<P 

y  and  determine  a  field  of  force.     The  length  of  F  may  now  be 

expressed   V \p2-\-\J/'2  and  the  projections  of  F  on  any  tangent  whose 
direction  is  y'  and  the  corresponding  normal  are 


Also  the  components  of  the  space  derivative  F  of  F  along  the  axes 
are  <p8  and  ^8  and  the  components  of  F  along  the  tangent  and  normal 

Hence  the  radius  of  curvature  for  conic  I  is  known  geometrically 
from  Property  3  in  terms  of  known  vectors. 


CONVERSION    OF    PROPERTIES. 


31 


The  radius  is 


To 


3(^4-/*)      (14-/2)3/2 


■  y  ^s 


Vi+/2 
10 


3 /'(«*+/*)- 


V)-Pi(l +/')"! 


The  general  conic  (36)  corresponding  to  the  typical  fourth  order 


equation  (32)  has  a  radius  of  curvature 


(1+/2)3-'2 


If  the 


2(3  A  y"-5)  y"' 
curves  of  5  have  Property  3  this  radius  equals  that  given  above. 


(1+/2)3 


V 


2  (3.4  y"-5)  y" 
Solving  for  A  we  have 
(52)  .4  = 


l  +  v/2["3v" 
10     "[ 


(v+y'tf-Pxil+y*) 


Piy" 


5  {<p+y'<p) 


3?'{iP+y'*)-Pi(\+yJi) 


Hence  a  set  of  °o  4  curves  having  Properties  1,  2,  3  are  curves 
in  a  field  of  force  and  are  represented  by  the  equation 


(53)     ^  = 


5  (v+y'v) 


v 


'  +  B  y'"+C 


3y"(<P+y'f)-Pi(i+y'2) 

where  P]  is  the  known  function  of  x,  y,  y'  given  in  (30),  B  is  an 
arbitrary  function  of  ,x,  y,  y',  y"  and  C  is  related  to  B  according 
to  (50). 

The  relation  (50)  may  now  be  modified  by  using  (52)  to  elim- 
inate A. 

4.  Converse  of  Property  4. 

The  most  general  equation  of  a  cubic  which  passes  through 
the  origin  once,  has  an  asymptote  parallel  to  the  *-axis  and  has 
a  double  point  on  the  asymptote  at  infinity  is  the  following : 

y*+A,  *  (y-4o)+43y2+44y  =  o. 

A  corresponding  equation  through  any  point  (v,  y)  as  origin  and 
having  the  asymptote  parallel  to  the  direction  y'  is 

(54)  (y2-y.'V2)'!+51(,v,+-y'y2)  (y2-y'.v2- B2) 

4-  B3(y2  -  y%)  *+B4(y2  -  y'x2)  =  o 

where  ;v2,  y*  are  referred  to  the  fixed  point  (x,  y)  as  origin,  y'  is  any 
fixed  direction  and  Pi,  P2,  P.s,  BA  are  arbitrary  functions  of  x,  y,  y'. 


32  CURVES  OF  CONSTANT  PRESSURE. 

Property  4  requires  that  such  a  cubic  shall  be  the  locus  of  the 
second  center  of  curvature  for  conic  I  as  the  direction  of  the 
element  y'  varies.  According  to  formulas  (40)  the  coordinates  of 
the  second  center  of  curvature  for  conic  (36)  are 

....                             -D(40yy'P,+3BZ)) 
(55)  Xi  =  


yi  = 


100  y"2Pi2 
D  f  10  ZV'(1  -3  y'2)  -3BD  y'\ 


100  y"2Pi2 
from  which  we  get 

d  (1+y2) 

y'2—yx2  = 


x2+y'y2  = 


10  y"Px 

-3D  (14-/2)  (10  Piy'y"+B  D) 


100  y"2Pi2 

Substituting  these  values  in  equation  (54)  and  simplifying, 
D2(14-/2)2-3  BxD  (1+/2)  (10  Piy'y"+B  D) 
4-30  BxB2Pxy"(lO  Piy'y"+B  D) 
4-10  £8(1+/2)  Piy"D+l00  BW'2PS  =  o. 

From  this  B  can  be  expressed  in  terms  of  Bu  B2,  B3,  B.u 

By  substituting  for  B  in  (53)  we  obtain  the  equation  of  a  set  of 

curves  having  Properties  1,  2,  3,  4. 

(56)    y"=. 5  (*+3W  ?"" 

j 2! I  z)2(i 4-/2)2 

^3  BXD  [D  (1+/2)  _  io  p^o/'jL     v  ^  y 

-30  ^Z)  Pi/y'(l+/2) 4-300  BxB.Px^y'y'"1 

4-10  5SP,D  /'(1 4-/2) 4- 100  £4/'2Pi2l 

4-C. 

In  this  form  Pi  and  D  are  defined  by  (30),  Blt  B2,  B3,  £4  are 
undetermined  functions  of  x,  y,  y'  and  C  is  related  to  the  other 
quantities  by  relation  (50)  which  may  now  be  rewritten  sub- 
stituting the  value  of  B. 

5.  Converse  of  Property  5. 

As  the  general  cubic  (54)  is  written  it  is  evident  that  B2 
represents  the  intercept  of  the  asvmptote  on  the  y2-axis.      Its 

B2 


intercept  on  the  normal  to  the  direction  y'  is  then 


Vl+/2 


CONVERSION    OF    PROPERTIES.  33 

According  to  Property  5  this  intercept  must  be  three-tenths  the 
ratio  of  the  tangential  component  of  the  force  to  the  normal  com : 
ponent  of  the  space  derivative  of  the  force.      Hence 

(57)  -      B2  = ^ . 

Substituting  this  value  of  B2  in  (56),  the  equation  of  curves  having 
Properties  1,  2,  3,  4,  5  is 


(58)    y 


5  (y+yV)  /"' 
3  y"(„+/f)_p1(i+/*) 

v'" 


+30  £3Pi(l+;/2)  (^+rV)  +  100  £4Prj 
4-/'[-6Pi(l+/2)3(^+yV)+30SiP1y(14-y2)2 

-io^PxMi+Z2)2] 

+Pl2(l+;/2)4} 
+c, 

where   the  only  unknown   coefficients  are  B\,   B9,   BA   functions 
of  x,  y,  y'. 

The  relation  (50)  of  C  to  B  is  further  specialized  by  the  new 
value  of  B. 

6.  Outline  of  the  final  steps  of  proof. 

It  remains  to  show  that  the  coefficients  B\,  B3,  BA  are  deter- 
mined by  Properties  6,  7,  8  so  that  the  equation  (32)  is  finally 
reduced  to  the  form  (31)  representing  curves  of  constant  pressure 
in  the  field  of  force  (<p,  \J/) .  It  will  be  seen  that  BA  and  B3  are  found 
in  terms  of  Bx  and  finally  Bi  in  terms  of  known  quantities.  After 
each  step  a  substitution  can  be  made  in  (58)  eliminating  one  of 
the  unknown  coefficients  but  the  resulting  expressions  are  long  and 
the  actual  writing  of  the  modified  equation  will  be  postponed  until 
the  characterization  of  the  curves  of  5  is  complete. 

7.  Converse  of  Property  6. 

In  order  to  impose  the  requirement  of  Property  6  upon  the 
set  of  curves  represented  by  (58)  the  cubic  (54)  at  the  point  of 
the  element  to  which  it  belongs  must  make  with  the  normal  to  the 
element  the  angle  described  in  Theorem  XIV.  The  tangent  to 
the  cubic  (54)  at  the  fixed  point  is  found  to  be 
y2       B&Ary'B* 


x2     -BxB«y'A-B; 


The  tangent  of  the  angle  which  this  line  makes 


34  CURVES  OF  CONSTANT  PRESSURE. 

D 

with  the  normal  is  —5-^-.     Replacing  Bt  bv  its  value  from  (57) 

and  setting  this  expression  equal  to  the  tangent  of  the  angle 
described  in  Theorem  XIV,  we  find  B±  in  terms  of  Bx  by  solving  the 
resulting  equation 

100  P,3       [  -P^^-^V)  J- 

The  substitution  of  (59)  in  (58)  produces  the  equation  of 
curves  having  Properties  1,  2,  3,  4,  5,  6.     Call  that  equation  (60). 

8.  Converse  of  Property  7. 

Following  the  method  employed  in  Article  10,  Chapter  III,  it 
is  found  that  two  curves  of  equation  (60)  having  a  given  curvature 
element  have  contact  of  the  fourth  order  with  their  osculating 
parabolas.     The  locus  of  the  foci  of  the  hyperosculating  parabolas  is 

(61)  4   P12(l4->^)2(a24-/32)2(«4-/3  y') 

-30  ^P^l+y'2)    (a24-£2)  (a+By'V 
4-6P,2£1(14-y'2)  (a24-/32)  03-a/)2 
-4P,(l  +  y'2)  [3  (1+y2)  (^4-vV)+5  £3Pi]  (a2+£2)  (jS-a/) 

(«+/3  /) 
4- [9   (14-/2)2(^+yV)2+30   ^Pjd+y'2)    (*+yty) 
4-100  £4  Pi2]  ()8-a  y')2(a+(3  3;') 

-9  PiE^l+y2)  (<^4-yV)  (<3-ay')3  =  o,  where  £4  is  written 
for  brevity  instead  of  (59).  The  form  of  this  equation  shows  that 
every  set  of  00 4  curves  having  Properties  1,2,3,  4,  5,  6  has  also  the 
property  that  the  foci  of  the  hyperosculating  parabolas  correspond- 
ing to  a  given  lineal  element  lie  on  a  quintic  curve  which  has  a 
triple  point  at  the  origin  and  has  the  element  tangent  as  a  double 
tangent. 

Consider  now  the  equation  of  the  tangent  to  the  third  branch 
of  (61)  at  the  origin 

(62)  J  -9  (14-y2)2(^  +  jV')2-30  B«Pi{l+/«)   (^4-yV) 
£_  1  -100  B4P{--9  ff1P1y'(l+;/2)   (y+yY) 

a      /  9  /(l+/2)2(^+/,A)2+30  #3P,(1+/2)  /(*>+/*) 
I  4-100  BiPfy'-9  BiPrd+y'2)  (*>+/£) 

The  tangent  of  the  angle  which  this  line  makes  with  the  element 
y'is 

(63)  9(14-/2)2(^4-yV)2+30^Pi(l+y/2)  (^4-/^)4-100  B4PX2 

9£1Pi(14-ri2)  (y+yV) 


CONVERSION    OF    PROPERTIES.  35 

To  satisfy  Property  7  this  must  equal  the  tangent  of  the  angle 

§ j 

which  the  element  makes  with  the  line  of  force,  —        '  ... 

When  this  equation  is  formed  and  solved  for  B3,  using  expres- 
sion (59)  for  B4,  we  have 


(64)     Ba  =  - 


10  Pi2 


Pi(l+/2)  (?+vty)+3  5,P,(^+vty) 
+B1P1P2 


Substituting  this  in  the  fourth  order  equation  (60)  as  already 
modified  by  Properties  1,  2,  3.  4,  5,  6  we  shall  have  (65)  as  the 
equation  of  curves  having  all  seven  properties  so  far  employed  in 
the  converse  problem. 

9.  Converse  of  Property  8. 

The  point  of  intersection  of  the  quintic  (61)  with  the  element 

y'  is  found  by  substituting  8  — a  y'  —  o  in  equation  (61).     The  co- 

r  .1  •  15  Pi  _,  0  15  Biy' 

ordmates  of  the  point  are  a  =  ^   ,,   — rr.  and  8  =  -   ,,  ,    ,„.    and 

2   (1+V2)  2(1+/2) 

its  distance  from  the  fixed  point  of  the  element  Va2+/32  =    -  ,  -. 

2Vl+/2 

By  Property  8  this  distance  must  be  five-halves  the  ratio  of  the 
normal  component  of  the  vector  F  to  the  normal  component  of  the 

vector  F  for  those  curves  which  are  hyperosculated  by  the  par- 
ticular parabola  whose  focus  is  the  intersection  of  the  quintic  and 
the  element  tangent.  Expressed  in  symbols  this  property  gives 
the  equation, 

15  Bl  5  P,  Vl-f-/2 

-^— .     Hence 


2  Vl-f/2  2  P3 

Pi(l  +  /2) 


(66)  Pi 


2>Pz 


Now  substituting  (66)  in  (65)  and  simplifying,  the  fourth  order 
equation  takes  the  form  of  equation  (31)  representing  all  the 
curves  of  constant  pressure  in  the  field  (<p,  \p). 

It  has  now  been  proved  that  the  eight  properties  derived  in 
Chapter  III  are  sufficient  to  identify  a  set  of  oo 4  curves  in  a  plane 
as  curves  of  constant  pressure  in  a  field  of  force. 


VITA 

Eugenie  M.  Morenus,  born  in  Cleveland,  New  York,  February  21, 

1881. 
Bachelor  of  Arts,  Vassar  College,  1904. 
Master  of  Arts,  Vassar  College,  1905. 
Graduate  Student,  University  of  Chicago,  University  of  Gottingen, 

Columbia  University. 
Teacher  in  Watertown  High  School,  January,  1906,  to  June,  1907. 
Substitute  in  Mathematics,  Vassar,  1907-1908. 
Teacher  in  Poughkeepsie  High  School,   1908-1909. 
Instructor  in  Mathematics  and  Latin,  Sweet  Briar  College,  1909- 

1916. 
Associate  Professor  of  Mathematics,  Sweet  Briar  College,  1916- 

1918. 

Professor  of  Mathematics,  Sweet  Briar  College,  1918 

Master's  Thesis:    "Some  Curves  Connected  with   a   System   of 
Similar  Conies." 


r. 


478713 


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